Question

examine this figure. which two pieces of information, if true, would help to prove that △lmp ≅ △nmp by hl? select two options.
□ point p is the midpoint of (overline{mk}).
□ line mk is the perpendicular bisector of (overline{ln}).
□ (overline{ml} cong overline{mp})
□ (overline{ml} cong overline{mn})
□ (overline{pk} cong overline{pk})

Explanation:

Step1: Recall HL Congruence

HL (Hypotenuse - Leg) congruence applies to right triangles. For \(\triangle LMP \cong \triangle NMP\) by HL, we need a right angle, hypotenuse congruence, and leg congruence.

Step2: Analyze Option 2

If Line \(MK\) is the perpendicular bisector of \(\overline{LN}\), then \(\angle LPM=\angle NPM = 90^\circ\) (right angles) and \(LP = NP\) (bisector). Also, \(MP\) is common.

Step3: Analyze Option 4

If \(\overline{ML}\cong\overline{MN}\), then \(ML = MN\) (hypotenuse), \(MP\) is common (leg), and \(\angle LPM=\angle NPM = 90^\circ\) (from perpendicular bisector or right angle). So with \(ML\cong MN\) and right angles, HL applies.

Answer:

B. Line \(MK\) is the perpendicular bisector of \(\overline{LN}\)
D. \(\overline{ML}\cong\overline{MN}\)